How to create an anime eye
Use a sphere and a lattice modifier.


Add a sphere for the eye.
Add a lattice object.
Select the eye and then &#8679; Shift select the lattice.
Parent the eye to the lattice &#9096; CtrlP and choose Lattice Deform.
In Edit mode for the lattice, move its points on the axis you want the eye to shrink.


Now as you rotate the eye, it will stay deformed within the lattice.